I am interested in the following type of objects (I will describe them later), but because I am very new to the subject of differential geometry, and I just have a very basic notion of the concepts, I was wondering if they have a particular name, so I can learn more about them.
As far as I understand, the concept of a differential manifold "can be motivated" through the study of surfaces in 3 dimensional space. Later one can show that the graph of a function in $\mathbf{R}^3$ has (under certain conditions) the structure of a differential manifold. The type of object I am interested in studying is a kind of surface in which there is a region where the surface splits while " still being tangent". I think a two-dimensional example is clearer, in that case think of a smooth line that has a point in which it splits in two lines, but in such a way that the new lines have the same tangent where they split (after that the lines can continue indefinitely or end). I illustrate this schematically in the next image:
I know that this object can be studied as a subset of $\mathbf{R}^2$ or $\mathbf{R}^3$ with the standard tools of calculus, and I know that one can study this set as a topological set, but can I also be studied under differential geometry or a slight modification of it? Does this type of objects have a name?
I have come across the name of "manifold with singularities" but they usually talk about cone type singularities, 'X' type singularities or in general points in which no tangent space can be constructed. But keeping in mind the motivation of the object, I see no problem in defining a tangent line to the curve at each point or a tangent plane, etc. So I am interested to know if there is an object in a generalization of differential geometry that "splits" while still having a kind of tangent space in each point. This object would almost be a manifold except for the splitting part.
Thank you for reading and for your time.
In general they are called branched manifolds.
The special case of a 1-dimemsional branched manifold is called a train track.